I need to derive the solution for the differential equation analytically:

y'' + g(t,y(t)) = 0
y'(0) = z_o
y(0) = y_o

I know the solution is:

y(t) = y_o + z_ot - single integral from 0 to t of (t-s)g(s,y(s))ds

I believe I need to assume something about the solution being a function of e^at somehow due to no damping, but I'm not sure.

You don't need to assume that. But you do need to know how to convert a differential equation to an equivalent Volterra integral equation. Find a book on intro to integral equations. First note the transformational formula (for derivation and n'th case, see integral equation text):

[tex]\int_0^t \int_0^t f(t)dtdt=\int_0^t(t-s)f(s)ds[/tex]

Now let:

[tex]y''=-f(t,y)[/tex]

and integrate both sides from 0 to t:

[tex]\int_0^t y'' dt=-\int_0^t f(t,y)dt[/tex]

[tex]y'(t)-z0=-\int_0^t f(t,y)dt[/tex]

Now integrate again and use the transformational expression to arive at the Volterra integral equation.

I just got the homework back. I used to volterra transformation but I was basically supposed to derive the transformation myself without just using it, which is pretty much what I expected. I had the prof explain to me how fundamentally it was possible to change a double integration into a single integration. He drew on the board on how to do it, change of variables using basically the technique I also found outlined here on wikipedia: